Branching rules, Kostka-Foulkes polynomials and $q$-multiplicities in tensor product for the root systems $B_{n},C_{n}$ and $D_{n}$

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Branching rules, Kostka-Foulkes polynomials and q-multiplicities in tensor product for the root systems

B _n, C _n and D_n

Cédric Lecouvey

To cite this version:

Cédric Lecouvey. Branching rules, Kostka-Foulkes polynomials and q-multiplicities in tensor product

for the root systems B_n, C_n and D_n. 2005. �hal-00003733v2�

ccsd-00003733, version 2 - 17 Jan 2005

Branching rules, Kostka-Foulkes polynomials and q-multiplicities in tensor product for the root systems B n , C n and D n

C´edric Lecouvey [email protected]

Abstract

The Kostka-Foulkes polynomials K _λ,µ ^φ (q) related to a root system φ can be defined as al- ternated sums running over the Weyl group associated to φ. By restricting these sums over the elements of the symmetric group when φ is of type B n , C n or D n , we obtain again a class K e _λ,µ ^φ (q) of Kostka-Foulkes polynomials. When φ is of type C n or D n there exists a duality beetween these polynomials and some natural q-multiplicities u λ,µ (q) and U λ,µ (q) in tensor product [14].

In this paper we first establish identities for the K e _λ,µ ^φ (q) which implies in particular that they can be decomposed as sums of Kostka-Foulkes polynomials K _λ,µ ^A

(q) with nonnegative integer coefficients. Moreover these coefficients are branching rule coefficients. This allows us to clarify the connection beetween the q-multiplicities u λ,µ (q), U λ,µ (q) and the polynomials K _λ,µ ^♦ (q) de- fined in [27]. Finally we show that u λ,µ (q) and U λ,µ (q) coincide up to a power of q with the one dimension sum introduced in [4] when all the parts of µ are equal to 1 which partially proves some conjectures of [14] and [27].

1 Introduction

Consider λ and µ two partitions of the set P n of partitions with n parts. The Schur-Weyl duality establishes that the dimension K _λ,µ ^A

of the weight space µ in the finite dimensional irreducible sl _n -module V ^A

(λ) of highest weight λ is equal to the multiplicity of V ^A

(λ) in the tensor product

V _µ ^A

= V ^A

(µ ₁ Λ 1 ) ⊗ · · · ⊗ V ^A

(µ _n Λ 1 ).

It follows from the Weyl character formula that K _λ,µ ^A

= P

σ∈S

(−1) ^l(σ) P ^A

(σ(λ + ρ) − (µ + ρ)) where P ^A

is the Kostant partition function which counts, in the root system of type A _n−1 , the number of decomposition of β ∈ Z ⁿ as a sum of positive roots. The Kostka-Foulkes polynomials can be defined by setting K _λ,µ ^A

(q) = P

σ∈S

(−1) ^l(σ) P _q ^A

(σ(λ + ρ) − (µ + ρ)) where P q ^A

is the q-Kostant partition function characterized by

Y

α positive root

1

(1 − qe ^α ) = X

β∈ Z

P _q ^A

(β)e ^β

with ρ = (n −1, ..., 0) the half sum of the positive roots. One can prove that they are the coefficients of the expansion of the Schur function s _µ (x) on the basis of Hall polynomials {P λ (x, q), λ ∈ P n } (see [19]). Then it follows from the theory of affine Hecke algebras that the Kostka-Foulkes polynomi- als are Kazhdan-Lusztig polynomials [18]. In particular they have nonnegative integer coefficients.

As proved by Lascoux and Sch¨ utzenberger this positivity result can also be obtained by using the

charge statistic ch on semistandard tableaux. More precisely we have K _λ,µ ^A

(q) = P

T ∈ST (λ)

q ^{ch(T )} where ST (λ) _µ is the set of semistandard tableaux of shape λ and weight µ. In [20], Nakayashiki and Yamada have shown that the charge can be computed from the combinatorial R-matrix corre- sponding to Kashiwara’s crystals associated to some U _q ( sl c _n )-modules.

Now consider φ a root system of type B _n , C _n or D _n . Write g _φ for the corresponding simple Lie algebra. The Kostka-Foulkes polynomials K _λ,µ ^φ (q) associated to φ are defined by setting

K _λ,µ ^φ (q) = X

w∈W

(−1) ^l(w) P _q ^φ (w(λ + ρ _φ ) − (µ + ρ _φ ))

where W _φ , ρ _φ and P q ^φ are respectively the Weyl group, the half sum of the positive roots and the q-partition function corresponding to φ. The polynomial K _λ,µ ^φ (q) can be considered as a q-analogue of the dimension of the weight space µ in V ^φ (λ). As Kazhdan-Lusztig polynomials, they have also nonnegative coefficients. In [16], we have obtained for the root systems B n , C n and D n a statistic on Kashiwara-Nakashima’s tableaux from which it is possible to deduce this positivity for particular pairs of partitions (λ, µ). Nevertheless as far as the author is aware, no combinatorial proof of this positivity result is known in general.

The Kostka-Foulkes polynomials K _λ,µ ^φ (q) cannot be directly interpreted as q-multiplicities in tensor products. So there does not exist an equivalent result to the Schur Weyl duality for the root system φ. Denote by V ^φ (λ) the finite dimensional irreducible g _φ -module of highest weight λ. In [14], we have introduced from determinantal expressions of the Schur functions associated to φ, two polynomials u _λ,µ (q) and U _λ,µ (q) which can be respectively regarded as quantizations of the multiplicities of V ^φ (λ) in the tensor products

V _µ ^φ = V ^φ (µ ₁ Λ ₁ ) ⊗ · · · ⊗ V ^φ (µ _n Λ ₁ ) and W _µ ^φ = W ^φ (µ ₁ Λ ₁ ) ⊗ · · · ⊗ W ^φ (µ _n Λ ₁ )

where for any i = 1, ..., n, W ^φ (µ _i Λ ₁ ) = V ^φ (µ _i Λ ₁ ) ⊕ V ^φ ((µ _i − 2)Λ ₁ ) ⊕ · · · ⊕ V ^φ ((µ _i mod2)Λ ₁ ). When n is sufficiently large they do not depend on the root system φ considered and we have established a duality result between the q-multiplicities u _λ,µ (q), U _λ,µ (q) and the polynomials

K e _λ,µ ^φ (q) = X

σ∈S

(−1) ^l(σ) P _q ^φ (w(λ + ρ _n ) − (µ + ρ _n ))

where ρ _n = (n, ..., 1). These polynomials K e _λ,µ ^φ (q) are also Kostka-Foulkes polynomials. So this result can be interpreted as a duality between q-analogues of weight multiplicities and q-analogues of tensor product multiplicities for the root systems B _n , C _n and D _n .

At the same time Shimozono and Zabrocki [27] have independently defined by using creating oper-

ators some polynomials K _λ,R ^♦ (q) where R is a sequence of rectangular partitions and ♦ a partition

of the set {∅, (1), (11), (2)}. These polynomials can also be regarded as q-multiplicities in tensor

products. In [4], Hatayama, Kuniba, Okado and Takagi have introduced for type C _n a quantization

X _λ,µ (q) of the multiplicity of V ^C

(λ) in W _µ ^C

. This quantization is based on the determination of

the combinatorial R-matrix of some U _q ^′ ( sp d 2n )-crystals in the spirit of [20]. It can be regarded as

a one dimension sum for the affine root system C n ⁽¹⁾ . In [14] and [27], the authors conjecture that

the polynomials X _λ,µ (q), U _λ,µ (q) and K _λ,µ ⁽²⁾ (q) coincide up to simple renormalizations. As observed

in [27], this conjecture can be related to the X = M conjecture which gives fermionic formulas for

the one dimension sum X. Note that the X = M conjecture have been proved in various cases for

all nonexceptional affine types [22], [23] and [24].

In this article we first obtained identities for the polynomials K e _λ,µ ^φ (q) which imply that they can be decomposed as sums of polynomials K _λ,µ ^A

(q). Moreover the coefficients of these decompo- sitions can be simply expressed in terms of branching rules coefficients. This gives in particular an elementary proof of the positivity of the Kostka-Foulkes polynomials K e _λ,µ ^φ (q). Next we obtain similar decompositions for the polynomials u _λ,µ (q) and U _λ,µ (q). By comparing these identities with those obtained for the polynomials K _λ,R ^♦ (q) in [27], we derive the equalities K _λ,µ ^(1,1) (q) = u _λ,µ (q ² ) and K _λ,µ ⁽²⁾ (q) = U _λ,µ (q ² ). Finally we establish some conjectures of [14] and [27] when all the parts of µ are equal to 1 (i.e. for the q-multiplicities defined in the tensor powers of the vector representation), namely we have

K _λ,(1 ^(1,1)

) (q) = u _λ,(1

₎ (q ² ) = q ^n−|λ| X _λ,(1

₎ (q ² ) and K _λ,(1 ⁽²⁾

) (q) = U _λ,(1

₎ (q ² ) = q ^2(n−|λ|) X _λ,(1

₎ (q ² ).

(1) In Section 2 we review some material on root systems, branching rules coefficients, Kostka- Foulkes polynomials and q-multiplicities u _λ,µ (q), U _λ,µ (q) we need in the sequel. In section 3 we obtain identities for the polynomials K e _λ,µ ^φ (q), u _λ,µ (q) and U _λ,µ (q) from which we clarify the relations between u _λ,µ (q), U _λ,µ (q) and K _λ,µ ^(1,1) (q), K _λ,µ ⁽²⁾ (q). Section 4 is devoted to the proof of (1). Note that the X = M conjecture is in particular true when all the parts of µ are equal to 1 [22]. Thus in this case the one dimension sums X and their corresponding fermionic formulas M are, up to simple renormalizations, Kazhdan-Lusztig polynomials.

Notation: In the sequel we frequently define similar objects for the root systems B _n C _n and D _n . When they are related to type B _n (resp. C _n , D _n ), we implicitly attach to them the label B (resp.

the labels C, D). To avoid cumbersome repetitions, we sometimes omit the labels B, C and D when our definitions or statements are identical for the three root systems.

2 Background

2.1 Convention for the root systems of types B _n , C _n and D _n

Consider an integer n ≥ 1. The weight lattice for the root system C n (resp. B n and D n ) can be identified with P _C

= Z ⁿ (resp. P _B

= P _D

=

Z 2

n

) equipped with the orthonormal basis ε _i , i = 1, ..., n. We take for the simple roots

 



α ^B _n

= ε _n and α ^B _i

= ε _i − ε _i+1 , i = 1, ..., n − 1 for the root system B _n α ^C _n

= 2ε n and α ^C _i

= ε i − ε i+1 , i = 1, ..., n − 1 for the root system C n

α ^D _n

= ε _n + ε _n−1 and α ^D _i

= ε _i − ε _i+1 , i = 1, ..., n − 1 for the root system D _n

. (2)

Then the set of positive roots are

 



R _B ⁺

= {ε _i − ε _j , ε _i + ε _j with 1 ≤ i < j ≤ n} ∪ {ε _i with 1 ≤ i ≤ n} for the root system B _n R _C ⁺

= {ε i − ε j , ε i + ε j with 1 ≤ i < j ≤ n} ∪ {2ε i with 1 ≤ i ≤ n} for the root system C n

R _D ⁺

n

= {ε i − ε _j , ε _i + ε _j with 1 ≤ i < j ≤ n} for the root system D _n

.

Denote respectively by P _B ⁺

n

, P _C ⁺

n

and P _D ⁺

n

the sets of dominant weights of so _2n+1 , sp _2n and so _2n .

Let λ = (λ ₁ , ..., λ _n ) be a partition with n parts. We will classically identify λ with the dominant weight P n

i=1 λ _i ε _i . Note that there exists dominant weights associated to the orthogonal root systems whose coordinates on the basis ε _i , i = 1, ..., n are not positive integers (hence which cannot be regarded as partitions). For each root system of type B _n , C _n or D _n , the set of weights having nonnegative integer coordinates on the basis ε ₁ , ..., ε _n can be identify with the set P n of partitions of length n. For any partition λ, the weights of the finite dimensional so _2n+1 , sp _2n or so _2n -module of highest weight λ are all in Z ⁿ . For any α ∈ Z ⁿ we write |α| = α ₁ +···+α n and kαk = P n−1

i=1 (n−i)α i . The conjugate partition of the partition λ is denoted λ ^′ as usual. Consider λ, µ two partitions of length n and set m = max(λ ₁ , µ ₁ ). Then by adding to λ ^′ and µ ^′ the required numbers of parts 0 we will consider them as partitions of length m.

The Weyl group W _B

= W _C

of so _2n+1 and sp _2n is identified to the subgroup of the permutation group of the set {n, ..., 2, 1, 1, 2, ..., n} generated by s _i = (i, i + 1)(i, i + 1), i = 1, ..., n − 1 and s _n = (n, n) where for a 6= b (a, b) is the simple transposition which switches a and b. We denote by l _B the length function corresponding to the set of generators s _i , i = 1, ...n.

The Weyl group W _D

of so _2n is identified to the subgroup of W _B

generated by the transpositions s _i = (i, i + 1)(i, i + 1), i = 1, ..., n − 1 and s ^′ _n = (n, n − 1)(n − 1, n). We denote by l _D the length function corresponding to the set of generators s ^′ _n and s _i , i = 1, ...n − 1.

Note that W _D

⊂ W _B

and any w ∈ W _B

verifies w(i) = w(i) for i ∈ {1, ..., n}. The action of w on β = (β ₁ , ..., β _n ) ∈ Z ⁿ is given by

w · (β ₁ , ..., β _n ) = (β ^w ₁ , ..., β ^w _n ) where β ^w _i = β _w(i) if σ(i) ∈ {1, ..., n} and β ^w _i = −β _w(i) otherwise.

The half sums ρ _B

, ρ _C

and ρ _D

of the positive roots associated to each root system B _n , C _n and D _n verify:

ρ _B

= (n − 1 2 , n − 3

2 , ..., 1

2 ), ρ _C

= (n, n − 1, ..., 1) and ρ _B

= (n − 1, n − 2, ..., 0).

In the sequel we identify the symmetric group S _n (which is the Weyl group of the root system A _n−1 ) with the subgroup of W _B

or W _D

generated by the s _i ’s, i = 1, ..., n − 1.

2.2 Branching rules coefficients

For any partition λ, we denote by V _n ^B (λ), V _n ^C (λ), and V _n ^D (λ) the finite dimensional irreducible modules of highest weight λ respectively for sp _2n , so _2n+1 and so _2n . Then V _n ^B (λ), V _n ^C (λ), and V _n ^D (λ) can also be regarded as irreducible representations respectively of the groups Sp _2n , So _2n+1 and So _2n . By restriction to GL _n , they decompose in a direct sum of irreducible rational representations. Recall that the irreducible rational representations of GL _n are indexed by the n-tuples

(γ ⁺ , γ ⁻ ) = (γ ⁺ ₁ , γ ⁺ ₂ , ..., γ ⁺ _p , 0, ..., 0, −γ ⁻ _q , ..., −γ ⁻ ₁ ) (3) where γ ⁺ and γ ⁻ are partitions of length p and q such that p + q ≤ n. Write V _n ^A (γ ⁺ , γ ⁻ ) for the irreducible rational representations of GL n of highest weight (γ ⁺ , γ ⁻ ). When γ ⁻ = ∅, we write simply V _n ^A (γ) instead of V _n ^A (γ ⁺ , γ ⁻ ).

As customary, we use for a basis of the group algebra Z [ Z ⁿ ], the formal exponentials (e ^β ) _β∈ Z

satisfying the relations e ^β

e ^β

= e ^β

^+β

. We furthermore introduce n independent indeterminates

x ₁ , ..., x _n in order to identify Z [ Z ⁿ ] with the ring of polynomials Z [x ₁ , ..., x _n , x ⁻¹ ₁ , ..., x ⁻¹ _n ] by writing e ^β = x ^β ₁

· · · x ^β _n

= x ^β for any β = (β ₁ , ..., β _n ) ∈ Z ⁿ .

Set

Y

1≤r<s≤n

1 − 1

x r x s

−1 Y

1≤i≤n

1 − 1

x i

−1

= X

β∈L

b(β )x ^−β , (4)

Y

1≤r≤s≤n

1 − 1

x _r x _s −1

= X

β∈L

c(β)x ^−β , Y

1≤r<s≤n

1 − 1

x _r x _s −1

= X

β∈L

d(β)x ^−β

where

L _B = {β ∈ Z ⁿ , β = X

1≤r<s≤n

e _r,s (ε _r + ε _s ) + X

1≤i≤n

e _i ε _i with e _r,s ≥ 0 and e _i ≥ 0}

L _C = {β ∈ Z ⁿ , β = X

1≤r≤s≤n

e _r,s (ε _r + ε _s ) with e _r,s ≥ 0} and L _D = {β ∈ Z ⁿ , β = X

1≤r<s≤n

e _r,s (ε _r + ε _s ) with e _r,s ≥ 0}.

Denote respectively by [V _n ^A (γ ⁺ , γ ⁻ ) : V _n ^B (λ)], [V _n ^A (γ ⁺ , γ ⁻ ) : V _n ^C (λ)] and [V _n ^A (γ ⁺ , γ ⁻ ) : V _n ^D (λ)], the multiplicities of V _n ^A (γ ⁺ , γ ⁻ ) in the restrictions of V _n ^B (λ), V _n ^C (λ) and V _n ^D (λ) to GL n .

Proposition 2.2.1 With the above notation, we have:

1. [V _n ^A (γ ⁺ , γ ⁻ ) : V _n ^B (λ)] = P

w∈W

(−1) ^l(w) b(w ◦ λ − (γ ⁺ , γ ⁻ )), 2. [V _n ^A (γ ⁺ , γ ⁻ ) : V _n ^C (λ)] = P

w∈W

(−1) ^l(w) c(w ◦ λ − (γ ⁺ , γ ⁻ )), 3. [V _n ^A (γ ⁺ , γ ⁻ ) : V _n ^D (λ)] = P

w∈W

(−1) ^l(w) d(w ◦ λ − (γ ⁺ , γ ⁻ )).

Proof. The proposition can be considered as a corollary of Theorem 8.2.1 of [3] with G one of the Lie groups So _2n+1 , Sp _2n , So _2n and H = GL _n .

For any partitions λ and ν of length n, write [V _n ^D (ν) : V _n ^B (λ)] for the multiplicity of V _n ^D (ν) in the restriction of V _n ^B (λ) to So _2n .

Lemma 2.2.2 With the notation above we have [V _n ^D (ν) : V _n ^B (λ)] = X

w∈W

(−1) ^l(w) 1 N (w ◦ λ − ν)

where for any β ∈ Z ⁿ , 1 N (β) = 1 if all the coordinates of β are nonnegative integers and 1 N (β) = 0

otherwise.

Proof. The lemma also follows from Theorem 8.2.1 of [3].

For any partitions λ and ν of length n, write [V _n ^B (λ) : V _2n ^A (ν)] for the multiplicity of V _n ^B (λ) in the restriction of V _2n ^A (ν) from GL _2n to So _2n . Similarly write [V _n ^C (λ) : V _2n ^A (ν)] for the multiplicity of V _n ^C (λ) in the restriction of V _2n ^A (ν ) from GL _2n to Sp _2n and [V _n ^D (λ) : V _2n ^A (ν)] for the multiplicity of V _n ^D (λ) in the restriction of V _2n ^A (ν) from GL _2n to So _2n+1 . Denote respectively by P n ⁽²⁾ and P n ^(1,1)

the sub-sets of P n containing the partitions with even rows and the partitions with even columns.

The Littlewood-Richardson coefficients are denoted c ^v _γ,λ as usual.

Let us recall a classical result by Littelwood (see [17] appendix p 295) Proposition 2.2.3 Consider λ and µ in P n . Then:

1. [V _n ^B (λ) : V _2n ^A (ν)] = P

γ∈P

c ^ν _γ,λ , 2. [V _n ^C (λ) : V _2n ^A (ν)] = P

γ∈P

c ^ν _γ,λ , 3. [V _n ^D (λ) : V _2n ^A (ν)] = P

γ∈P

c ^ν _γ,λ .

The proposition below follows immediately from Theorem A ₁ of [10].

Proposition 2.2.4 Consider ν ∈ P n and λ ⁺ , λ ⁻ two partitions such that (λ ⁺ , λ ⁻ ) has length n.

Then:

1. [V _n ^A (λ ⁺ , λ ⁻ ) : V _n ^B (ν)] = P

γ,δ∈P

c ^ν _γ,δ c ^δ _λ

,λ

, 2. [V _n ^A (λ ⁺ , λ ⁻ ) : V _n ^C (ν)] = P

γ,δ∈P

c ^ν _γ,δ c ^δ _λ

,λ

, 3. [V _n ^A (λ ⁺ , λ ⁻ ) : V _n ^D (ν)] = P

γ,δ∈P

c ^ν _γ,δ c ^δ _λ

,λ

.

When (λ ⁺ , λ ⁻ ) = λ is a partition (that is λ ⁻ = ∅), we obtain the following dualities:

Corollary 2.2.5 Consider λ, ν two partitions of length n, then 1. [V _n ^A (λ) : V _n ^B (ν)] = [V _n ^B (λ) : V _2n ^A (ν)] = P

γ∈P

c ^ν _γ,λ , 2. [V _n ^A (λ) : V _n ^C (ν)] = [V _n ^D (λ) : V _2n ^A (ν )] = P

γ∈P

c ^ν _γ,λ , 3. [V _n ^A (λ) : V _n ^D (ν )] = [V _n ^C (λ) : V _2n ^A (ν )] = P

γ∈P

c ^ν _γ,λ .

2.3 Kostka-Foulkes polynomials

For any w ∈ W _B

, the dot action of w on β ∈ Z ⁿ is defined by w ◦ β = w(β + ρ _B

) − ρ _B

.

The q-analogue P _q ^B

of the Kostant partition function corresponding to the root system B n is defined by the equality

Y

α∈R

1

1 − qx ^α = X

β∈ Z

P _q ^B

(β)x ^β .

Note that P _q ^B

(β ) = 0 if β is not a linear combination of positive roots of R ⁺ _B

n

with nonnegative coefficients. We write similarly P _q ^C

and P _q ^D

for the q-partition functions associated respectively to the root systems C _n and D _n . Given λ and µ two partitions of length n, the Kostka-Foulkes polynomials of types B _n , C _n and D _n are then respectively defined by

K _λ,µ ^B

(q) = X

σ∈W

(−1) ^l(σ) P _q ^B

(σ(λ + ρ _B

) − (µ + ρ _B

)), K _λ,µ ^C

(q) = X

σ∈W

(−1) ^l(σ) P _q ^C

(σ(λ + ρ _C

) − (µ + ρ _C

)), K _λ,µ ^D

(q) = X

σ∈W

(−1) ^l(σ) P _q ^D

(σ(λ + ρ _D

) − (µ + ρ _D

)).

Set

K e _λ,µ ^B

(q) = X

σ∈S

(−1) ^l(σ) P _q ^B

(σ(λ + ρ _B

) − (µ + ρ _n )), K e _λ,µ ^C

(q) = X

σ∈S

(−1) ^l(σ) P _q ^C

(σ(λ + ρ _C

) − (µ + ρ _n )), K e _λ,µ ^D

(q) = X

σ∈S

(−1) ^l(σ) P _q ^D

(σ(λ + ρ _D

) − (µ + ρ _n ))

where ρ _n = (n, ..., 1). In [14], we have proved that the polynomials K e _λ,µ (q) are also Kostka-Foulkes polynomials. More precisely we have:

Lemma 2.3.1 Consider λ, µ two partitions of length n such that |λ| ≥ |µ| . Let k be any integer such that k ≥ ^|λ|−|µ| ₂ . Then we have

K e _λ,µ (q) = K _λ+kκ

_,µ+kκ

(q) where κ n = (1, ..., 1) ∈ Z ⁿ .

Remark: Since σ(κ _n ) = κ _n for any σ ∈ S _n , we have K e _λ+kκ

_,µ+kκ

(q) = K e _λ,µ (q) for any integer k ≥ 0. So we can extend the above definition of K e _λ,µ (q) for λ and µ decreasing sequences of integers (positive or not).

2.4 The q-multiplicities u _λ,µ (q) and U _λ,µ (q) Set

Y

1≤i<j≤n

1 1 − q ^x _x

j

Y

1≤r<s≤n

1

1 − _x

^q _x

= X

β∈ Z

f _q (β)x ^β and Y

1≤i<j≤n

1 1 − q _x ^x

j

Y

1≤r≤s≤n

1 1 − _x ^q

i

x

= X

β∈ Z

F _q (β)x ^β .

Given λ and µ two partitions of length n, let u _λ,µ (q) and U _λ,µ (q) be the two polynomials defined

by

u _λ,µ (q) = X

σ∈S

(−1) ^l(σ) f _q (σ(λ + ρ _n ) − µ − ρ _n ) and U _λ,µ (q) = X

σ∈S

(−1) ^l(σ) F _q (σ(λ + ρ _n ) − µ − ρ _n ) where ρ _n = (n, ..., 1). Then u _λ,µ (q) and U _λ,µ (q) can be regarded as quantizations of tensor product multiplicities [14]. Consider the tensor products

V _µ ^B = V ^B (µ ₁ Λ 1 ) ⊗ · · · ⊗ V ^B (µ _n Λ 1 ), V _µ ^C = V ^C (µ ₁ Λ 1 ) ⊗ · · · ⊗ V ^C (µ _n Λ 1 ), V _µ ^C = V ^D (µ ₁ Λ ₁ ) ⊗ · · · ⊗ V ^D (µ _n Λ ₁ )

and

W _µ ^B = W ^B (µ ₁ Λ ₁ ) ⊗ · · · ⊗ W ^B (µ _n Λ ₁ ), W _µ ^C = W ^C (µ ₁ Λ ₁ ) ⊗ · · · ⊗ W ^C (µ _n Λ ₁ ), W _µ ^D = W ^D (µ ₁ Λ ₁ ) ⊗ · · · ⊗ W ^D (µ _n Λ ₁ )

where for any k ∈ N , W (k ₁ ) = V (kΛ ₁ ) ⊕ V ((k − 2)Λ ₁ ) ⊕ · · · ⊕ V ((k mod 2)Λ ₁ ). Then we have the following proposition:

Proposition 2.4.1 [14] Let λ and µ be two partitions of length n. Then

1. u _λ,µ (q) is a q-analogue of the multiplicity of the representation V (λ) in V _µ , 2. U _λ,µ (q) is a q-analogue of the multiplicity of the representation V (λ) in W µ . Remarks:

(i) : It follows from the definition of f _q and F _q that u _λ,µ (q) = U _λ,µ (q) = 0 if |λ| > |µ| .

(ii) : When q = 1, we recover that the multiplicities of V ^B (λ), V ^C (λ) and V ^D (λ) respectively in V _µ ^B , V _µ ^C , and V _µ ^D are equal [10].

(iii) : In [14], we have also obtained that u _λ,µ (q) and U _λ,µ (q) can be regarded as q-multiplicities in tensor product of column shaped representations.

(iv) : Like the definition of K e _λ,µ (q), the definitions of u _λ,µ (q) and U _λ,µ (q) can also be extended for λ and µ decreasing sequences of integers.

(v) : Consider λ, µ ∈ P n and set λ ^# = (λ ₁ , ..., λ _n , 0), µ ^# = (µ ₁ , ..., µ _n , 0). Then u _λ

,µ

(q) = u _λ,µ (q) and U _λ

,µ

(q) = U _λ,µ (q).

Theorem 2.4.2 [14] Consider λ, µ two partitions of length n and set m = max(λ ₁ , µ ₁ ). Then b λ = (m − λ _n , ..., m − λ ₁ ) and b µ = (m − µ _n , ..., m − µ ₁ ) are partitions of length n and

u _λ,µ (q) = K e _b ^D

λ,b µ (q), U _λ,µ (q) = K e _b ^C

λ,b µ (q).

Write I for the involution defined on Z ⁿ by I(β ₁ , ..., β _n ) = (−β _n , ..., −β ₁ ). For any decreasing sequence of integers γ, I (γ) is also a decreasing sequence of integers. By Theorem 2.4.2, this means that the correspondences

u _λ,µ (q) ←→ K e _I(λ),I(µ) ^D

(q) and U _λ,µ (q) ←→ K e _I(λ),I(µ) ^C

(q) (5)

where λ, µ are decreasing sequence of integers can be interpreted as a duality result for the q-

multiplicities associated to the classical root systems.

2.5 Crystals of type C _n and the one dimension sum X _λ,µ (q)

We have seen that U _λ,µ (q) can be regarded as a q-analogue of the multiplicity of V (λ) in W _µ ^C . In [4], Hatayama, Kuniba, Okado and Takagi have introduced another quantification X _λ,µ (q) of this multiplicity based on the determination of the combinatorial R-matrix of certain U _q ^′ (C n ⁽¹⁾ )-crystals B _k ^C . Considered as the crystal graph of a U _q (C _n )-module, B ^C _k is isomorphic to

B ^C (kΛ ₁ ) ⊕ B ^C ((k − 2)Λ ₁ ) ⊕ · · · ⊕ B ^C (k mod 2Λ ₁ ) (6) where for any i ∈ {k, k − 2, ..., k mod 2}, B ^C (kΛ ₁ ) is the crystal graph of the irreducible finite dimensional U _q (C _n )-module of highest weight kΛ ₁ . In [9], Kashiwara and Nakashima have obtained a natural labelling of the vertices of B ^C (kΛ ₁ ) by one-row tableaux of length k filled by letters of the alphabet

C n = {1 < · · · < n − 1 < n < n < n − 1 < · · · < 1}

such that the letters increase from left to right (that is by semistandard one-row tableaux on C n ).

Then the vertices of B _k ^C can be depicted by one-row tableaux of length k by adding p pairs (0, 0) to the tableaux appearing in the crystals B ^C ((k − 2p)Λ ₁ ) of the decomposition (6). Then by setting

C n+1 = {0 < 1 < · · · < n − 1 < n < n < n − 1 < · · · < 1 < 0}

the crystal B _k ^C can be regarded as a subcrystal of the C n+1 -crystal associated to the dominant weight kΛ ₁ (labelled by the one-row semistandard tableaux on C n+1 with length k).

Recall that the combinatorial R-matrix associated to crystals B _k ^C is equivalent to the description of the crystal graph isomorphisms

B _l ^C ⊗ B _k ^C → ^≃ B _k ^C ⊗ B _l ^C b ₁ ⊗ b ₂ 7−→ b ^′ ₂ ⊗ b ^′ ₁

together with the energy function H ^C on B _l ^C ⊗ B _k ^C . As proved in [4], this can done by using the insertion algorithm for C _n+1 -tableaux of [1] or [15]. In the sequel we only need the description of the energy function H ^C . Consider b ₁ ∈ B _l ^C and b ₂ ∈ B _k ^C and denote by z = min(#0 in b ₁ ,

#0 in b 2 ) = min(#0 in b 1 , #0 in b 2 ). Let b ^∗ ₁ and b ^∗ ₂ be the row tableaux obtained by erasing z pairs of letters (0, 0) in b ₁ and b ₂ . Write l ^∗ and k ^∗ for the lengths of b ^∗ ₁ and b ^∗ ₂ . Denote by P _n+1 ^C (b ^∗ ₁ ⊗ b ^∗ ₂ ) the tableau obtained by inserting the row b ^∗ ₂ into the row b ^∗ ₁ following the C _n+1 - insertion algorithm. Then P _n+1 ^C (b ^∗ ₁ ⊗ b ^∗ ₂ ) is a two-row C n+1 -tableau which contains k ^∗ + l ^∗ letters.

Since the plactic relations for the root system C _n+1 are not homogeneous, pairs of letters (0, 0) can appear in P _n+1 ^C (b ^∗ ₁ ⊗ b ^∗ ₂ ). Write m for the length of the shortest row of P _n+1 ^C (b ^∗ ₁ ⊗ b ^∗ ₂ ).

Proposition 2.5.1 [4] For any b ₁ ⊗ b ₂ in B _l ^C ⊗ B _k ^C we have H ^C (b ₁ ⊗ b ₂ ) = min(l ^∗ , k ^∗ ) − m.

The multiplicity of V (λ) in W _µ ^C is then equal to the number of highest weight vertices of weight λ in the crystal B _µ ^C = B _µ ^C

⊗ · · · ⊗ B _µ ^C

n

. Then the one dimension sum X _λ,µ (q) for C n ⁽¹⁾ -crystals is defined in [4] by

X _λ,µ (q) = X

b∈E

q ^H

^(b) with H ^C (b) = X

0≤i<j≤n

H ^C (b _i ⊗ b ⁽ⁱ⁺¹⁾ _j )

where E _λ is the set of highest weight vertices b = b ₁ ⊗ · · · ⊗ b _n in B _µ ^C of highest weight λ, b ⁽ⁱ⁾ _j is determined by the crystal isomorphism

B _µ ^C

i

⊗ B _µ ^C

⊗ B _µ ^C

⊗ · · · ⊗ B _µ ^C

j

→ B _µ ^C

i

⊗ B _µ ^C

j

⊗ B _µ ^C

· · · ⊗B _µ ^C

b _i ⊗ b _i+1 ⊗ · · · ⊗ b _j → b ⁽ⁱ⁾ _j ⊗ b ^′ _i ⊗ · · · ⊗ b ^′ _j−1 (7) and for any j = 1, ..., n, H(b ₀ ⊗ b ⁽¹⁾ _j ) is equal to the number of letters 0 in b ⁽¹⁾ _j .

When |λ| = |µ| the vertices of E _λ contain only unbarred letters. In this case H ^A (b _i ⊗ b ⁽ⁱ⁺¹⁾ _j ) = H ^C (b _i ⊗ b ⁽ⁱ⁺¹⁾ _j ) that is the energy function of type A defined on the vertices b which do not contain any barred letter is the restriction of that of type C. In [20] Nakayashiki and Yamada have proved the equality H ^A (b) = ch(Q(b)) where Q(b) is the semistandard tableau associated to b by generalizing the Robinson-Schensted correspondence and ch is the charge defined by Lascoux and Sch¨ utzenberger in [12] and [13]. Recall that the charge statistic verifies

K _λ,µ ^A

(q) = X

T ∈SST

(λ)

q ^{ch(T )} (8)

where SST _µ (λ) is the set of semistandard tableaux of shape λ and weight µ. This implies that X _λ,µ (q) = K _λ,µ ^A

(q) when |λ| = |µ| .

Many computations suggest the following identities:

Conjecture 2.5.2 Consider λ ∈ P n . Then we have

(i) : u _λ,(1

₎ (q) = q ^{|µ|−|λ| 2} X _λ,(1

₎ (q),

(ii) : U _λ,µ (q) = q ^|µ|−|λ| X _λ,µ (q) for any µ ∈ P _n . Remarks:

(i): A q-analogue for the multiplicity of V _n ^C (λ) in W ^C (µ) can also be defined from rigged config- urations. The X = M conjecture gives a simple relation called fermionic formula between X and M. In [22], Okado, Schilling and Shimozono have proved this conjecture when all the parts of µ are equal to 1. In 4, we will prove (i) of Conjecture 2.5.2 and (ii) when µ = (1, ..., 1). Thus by combining our results with those of [22] we obtain very simple relations between the three different q-analogues for the multiplicity of V _n ^C (λ) in W ^C (1 ⁿ ) = V (1 ⁿ ).

(ii): By Theorem 2.4.2, the polynomials U _λ,µ (q) are Kostka-Foulkes polynomials. Thus (ii) of Conjecture 2.5.2 implies that, up to a simple renormalization, the one dimension sums X _λ,µ (q) are affine Kazhdan-Lusztig polynomials.

3 Identities for the q-multiplicities

3.1 Decomposition of the K e _λ,µ (q) in terms of the K _λ,µ ^A

(q) Denote by P _q ^A the q-Kostant partition function defined by

Y

1≤i<j≤n

1 − q x _i x _j

−1

= X

η∈ Z

P _q ^A (η)x ^η

Then for any λ, µ ∈ P n , the Kostka-Foulkes polynomial K _λ,µ ^A

(q) is such that K _λ,µ ^A

(q) = X

σ∈S

(−1) ^l(σ) P _q ^A (σ ◦ λ − µ).

Since K _λ+kκ ^A

_,µ+kκ

(q) = K _λ,µ ^A

(q) for any positive integer k, the definition of K _λ,µ ^A

(q) can be extended for λ and µ decreasing sequence of integers. By definition of the q-partition function P _q ^C we must have

X

β∈ Z

P _q ^C (β)x ^β = Y

1≤r≤s≤n

(1 − qx r x s ) ⁻¹ Y

1≤i<j≤n

1 − q x i

x _j −1

.

Now we can write by (4)

Y

1≤r≤s≤n

(1 − qx _r x _s ) ⁻¹ = X

δ∈L

q ^|δ|/2 c(δ)x ^δ (9)

since the number of roots appearing in a decomposition of δ ∈ L _C as a sum of positive roots ε _r + ε _s with 1 ≤ r ≤ s ≤ n is always equal to |δ| /2. Thus we obtain

X

β∈ Z

P _q ^C (β)x ^β = X

η∈ Z

X

δ∈L

q ^|δ|/2 P _q ^A (η)c(δ)x ^δ+η . (10) By using similar arguments for P _q ^D we derive the following lemma:

Lemma 3.1.1 For any β ∈ Z ⁿ we have P _q ^C (β ) = X

δ∈L

q ^|δ|/2 c(δ)P _q ^A (β − δ) and P _q ^C (β) = X

δ∈L

q ^|δ|/2 d(δ)P _q ^A (β − δ)

where L ^|β| _C = {δ ∈ L C , |δ| = |β|} and L ^|β| _D = {δ ∈ L D , |δ| = |β|}.

Proof. >From (10) we derive the equality P _q ^C (β) = P

η+δ=β c(δ)q ^|δ|/2 P _q ^A

(η). Since P _q ^A

(η) = 0 when |η| 6= 0, we can suppose |η| = 0 and |δ| = |β| in the previous sum. Then δ ∈ L ^|β| _C and the result follows immediately. The proof for P _q ^D (β ) is similar.

Remark: A similar result for the q-partition function P _q ^B does not exit. Indeed the number of roots appearing in a decomposition of δ ∈ L B as a sum of positive roots ε r + ε s with 1 ≤ r < s ≤ n and ε _i with 1 ≤ i ≤ n does not depend only of |δ| since |ε r + ε _s | 6= |ε i | .

Proposition 3.1.2 Consider λ, µ ∈ P n such that |λ| ≥ |µ| . Then we have:

1. K e _λ,µ ^C

(q) = q ^{|λ|−|µ| 2} P

γ∈ P e

P

σ∈S

(−1) ^l(σ) c(σ ◦ λ − γ)K _γ,µ ^A

(q) 2. K e _λ,µ ^D

(q) = q ^{|λ|−|µ| 2} P

γ∈ P e

P

σ∈S

(−1) ^l(σ) d(σ ◦ λ − γ)K γ,µ ^A

(q)

where P e n = {γ = (γ ₁ , ..., γ _n ) ∈ Z ⁿ , γ ₁ ≥ γ ₂ ≥ · · · ≥ γ _n }.

Proof. We have

K e _λ,µ ^C

(q) = X

σ∈S

(−1) ^l(σ) P _q ^C

(σ(λ + ρ _n ) − (µ + ρ _n )).

Hence from the previous lemma we derive K e _λ,µ ^C

(q) = X

δ∈L

c(δ )q ^|δ|/2 X

σ∈S

(−1) ^l(σ) P _q ^A

(σ(λ + ρ _n ) − (µ + δ + ρ _n )) and

K e _λ,µ ^C

(q) = X

σ∈S

(−1) ^l(σ) X

δ∈L

c(δ)q ^|δ|/2 P _q ^A

(σ(λ + ρ _n − σ ⁻¹ (δ)) − (µ + ρ _n )) For any σ ∈ S n , we have σ ⁻¹ (L ^|β| _C ) = L ^|β| _C and c(δ ) = c(σ(δ)). Thus we obtain

K e _λ,µ ^C

(q) = X

σ∈S

(−1) ^l(σ) X

δ∈L

c(δ)q ^|δ|/2 P _q ^A

(σ(λ + ρ _n −δ) − (µ + ρ _n )) = X

δ∈L

c(δ)q

K _λ−δ,µ ^A

(q).

(11) Now K _λ−δ,µ ^A

(q) = 0 or there exits σ ∈ S n and γ ∈ P e n such that γ = σ ⁻¹ ◦ (λ − δ). It follows that

K e _λ,µ ^C

(q) = X

σ∈S

(−1) ^l(σ) X

γ∈ P e

c((λ + ρ) − σ(γ + ρ))K _γ,µ ^A

(q).

Since c(δ) = c(σ(δ)) for any σ ∈ S n , and δ ∈ L _C we obtain the desired equality K e _λ,µ ^C

(q) = X

γ∈ P e

X

σ∈S

(−1) ^l(σ) c(σ(λ + ρ) − (γ + ρ))K _γ,µ ^A

(q)

The proof is similar for K e _λ,µ ^D

(q).

By Lemma 2.3.1, K e _λ,µ ^C

(q) and K e _λ,µ ^D

(q) are Kazhdan-Lusztig polynomials. Thus they have nonneg- ative integer coefficients. This property can also be obtained from the proposition below:

Corollary 3.1.3 Consider λ, µ ∈ P n . For k a sufficiently large integer we have:

1. K e _λ,µ ^C

(q) = q ^{|λ|−|µ| 2} P

γ∈ P e

[V _n ^A (γ + kκ _n ), V _n ^C (λ + kκ _n )]K γ,µ ^A

(q), 2. K e _λ,µ ^D

(q) = q ^{|λ|−|µ| 2} P

γ∈ P e

[V _n ^A (γ + kκ _n ), V _n ^D (λ + kκ _n )]K γ,µ ^A

(q).

Proof. Consider γ ∈ P e n and write γ = (γ ⁺ , γ ⁻ ) as in (3). We have by Proposition 2.2.1 [V _n ^A (γ) : V _n ^C (λ)] = P

w∈W

(−1) ^l(w) c(w ◦ λ − γ). Now if k is sufficiently large γ + kκ n is a partition and [V _n ^A (γ) : V _n ^C (λ)] = P

w∈W

(−1) ^l(w) c(w ◦ λ − γ + w(kκ _n ) − kκ _n ). Suppose that w / ∈ S n , then |w(kκ n ) − kκ _n | ≤ −2k. Thus we can choose k such that γ + kκ _n is a partition and w ◦ λ − γ + w(kκ _n ) − kκ _n ∈ / L _C for any w ∈ W _C

such that w / ∈ S _n . For such an integer k we have c(w ◦ λ − γ + w(kκ _n ) − kκ _n ) = 0. Then we derive 1 from 1 of Proposition 3.1.2. We prove 2 by using similar arguments.

To obtain a decomposition of the polynomial K e _λ,µ ^B

(q) as a sum of polynomials K _γ,µ ^A

(q), we

need first to obtain its decomposition in terms of the polynomials K e _ν,µ ^D

(q).

Proposition 3.1.4 Consider λ, µ ∈ P n such that |λ| ≥ |µ|. Then for k a sufficiently large integer we have:

K e _λ,µ ^B

(q) = X

ν∈ P e

q ^|λ|−|ν| [V _n ^D (ν + kκ _n ), V _n ^B (λ + kκ _n )] K e _ν,µ ^D

(q).

Proof. By definition of the partition functions P _q ^B

and P _q ^D

we can write X

β∈ Z

P _q ^B

(β)x ^β = X

γ∈ Z

P _q ^D

(γ)x ^γ Y n k=1

1 1 − qx _k . We have Q _n

k=1

1

1 − qx _k = P

δ∈ N

q ^|δ| x ^δ . Thus we derive X

β∈ Z

P _q ^B

(β)x ^β = X

γ∈ Z

X

δ∈ N

q ^|δ| P _q ^D

(γ)x ^γ+δ .

This implies that P _q ^B

(β) = P

δ∈ N

q ^|δ| P _q ^D

(β − δ) and K e _λ,µ ^B

(q) = X

σ∈S

(−1) ^l(σ) X

δ∈ N

q ^|δ| P _q ^D

(σ(λ + ρ _n ) − (µ + δ + ρ _n )).

By using similar arguments to that of the proof of Proposition 3.1.2, we obtain K e _λ,µ ^B

(q) = q ^|λ|−|µ| X

ν∈ P e

X

σ∈S

(−1) ^l(σ) 1 N (σ(λ + ρ _n ) − (ν + ρ _n ))K _ν,µ ^D

(q).

There exists a sufficiently large integer k for which w(λ + ρ _n + kκ _n ) − (ν + ρ _n + kκ _n ) ∈ / N ⁿ for any w ∈ W _B

such that w / ∈ S n . Then it follows from Lemma 2.2.2 that [V _n ^D (ν + kκ _n ) : V _n ^B (λ + kκ _n )] = P

σ∈S

(−1) ^l(σ) 1 ^N (σ ◦ λ − ν) and the proposition is proved.

Remark: From 2 of Corollary 3.1.3 and the above proposition we obtain K e _λ,µ ^B

(q) = X

ν∈ P e

X

γ∈ P e

q

2

[V _n ^D (ν + kκ n ), V _n ^B (λ + kκ n )][V _n ^A (γ + kκ n ), V _n ^D (ν + kκ n )]K _γ,µ ^A

(q)

which implies in particular that the polynomials K e _λ,µ ^B

(q) have non negative coefficients.

3.2 Decomposition of u _λ,µ (q) and U _λ,µ (q) in terms of the K _λ,µ ^A

(q)

By using similar arguments to the proof of Lemma 3.1.1, we can establish that for any β ∈ Z ⁿ we have

f q (β) = X

δ∈L

q ^|δ|/2 d(δ)P _q ^A (β + δ) and F q (β) = X

δ∈L

q ^|δ|/2 c(δ)P _q ^A (β + δ) which implies the following proposition:

Proposition 3.2.1 Consider λ, µ ∈ P n such that |µ| ≤ |λ| . Then we have

1. U _λ,µ (q) = q ^{|µ|−|λ| 2} P

ν∈P

[V _n ^A (λ), V _n ^C (ν)]K _ν,µ ^A

(q), 2. u _λ,µ (q) = q ^{|µ|−|λ| 2} P

ν∈P

[V _n ^A (λ), V _n ^D (ν)]K ν,µ ^A

(q).

Proof. We proceed as in the proof of Proposition 3.1.2 and (11) is replaced by U _λ,µ (q) = X

δ∈L

c(δ)q

K _λ+δ,µ ^A

(q).

This time K _λ+δ,µ ^A

(q) = 0 or there exits σ ∈ S n and ν ∈ P n such that ν = σ ⁻¹ ◦ (λ + δ). Note that ν can not have negative coordinates for λ + δ have non negative coordinates. We deduce:

U _λ,µ (q) = q

X

σ∈S

(−1) ^l(σ) X

ν∈P

c(σ(ν + ρ _n ) − (λ + ρ _n ))K _γ,µ ^A

(q).

We have seen in the proof of Corollary 3.1.3 that for k a sufficiently large integer we have [V _n ^A (λ + kκ _n ), V _n ^C (ν + kκ _n )] = X

σ∈S

(−1) ^l(σ) c(σ ◦ ν − λ).

Since ν is a partition and c ^ν _γ,λ+kκ ^+kκ

n

= c ^ν _γ,λ for any k ∈ N , we deduce from Corollary 2.2.5 that [V _n ^A (λ), V _n ^C (ν )] = [V _n ^A (λ + kκ n ), V _n ^C (ν + kκ n )] which proves 1. Assertion 2 is obtained similarly.

3.3 Link with the polynomials K _λ,µ ^♦ ( q ) of Shimozono and Zabrocki We deduce from Corollary 2.2.5 and Proposition 3.2.1:

Theorem 3.3.1 Consider λ, µ ∈ P n such that |µ| ≥ |λ| Then we have the following equalities (i) : u _λ,µ (q) = q

X

ν∈P

[V _n ^C (λ), V _2n ^A (ν)]K _ν,µ ^A

(q) = q

X

ν∈P

[V _n ^A (λ), V _n ^D (ν )]K _ν,µ ^A

(q)

= q

X

ν∈P

X

γ∈P

c ^ν _γ,λ K _ν,µ ^A

(q),

(ii) : U _λ,µ (q) = q

X

ν∈P

[V _n ^D (λ), V _2n ^A (ν)]K _ν,µ ^A

(q) = q

X

ν∈P

[V _n ^A (λ), V _n ^C (ν)]K _ν,µ ^A

(q)

= q

X

ν∈P

X

γ∈P

c ^ν _γ,λ K _ν,µ ^A

(q).

By comparing the leftmost equalities of the above theorem with equality (7.6) of [27] we obtain

K _λ,µ ^(1,1) (q) = u _λ,µ (q ² ) and K _λ,µ ⁽²⁾ (q) = U _λ,µ (q ² ) (12)

where K _λ,µ ^(1,1) (q) and K _λ,µ ⁽²⁾ (q) are polynomials defined by Shimozono and Zabrocki by using creating

operators on formal series. In particular the polynomials K _λ,µ ^(1,1) (q) and K _λ,µ ⁽²⁾ (q) are Kazhdan-Lusztig

polynomials specialized at q ² .

Remark: In [27] the authors have also defined another polynomial denoted K _λ,µ ⁽¹⁾ (q) verifying K _λ,µ ⁽¹⁾ (q) = q ^|µ|−|λ| X

ν∈P

X

γ∈P

c ^ν _γ,λ K _ν,µ ^A

(q ² ). (13)

From the duality (5), it is tempting to introduce the polynomial V _λ,µ (q) = K e _I(λ),I(µ) (q). A similar result than Proposition 3.2.1 can not exist for V _λ,µ (q) (see the remark following Lemma 3.1.1).

Nevertheless, by using Corollary 2.2.5, one can establish that V _λ,µ (1) = X

ν∈P

[V _n ^B (λ), V _2n ^A (ν)]K _ν,µ ^A

(1) = X

ν∈P

X

γ∈P

c ^ν _γ,λ K _ν,µ ^A

(1).

Thus K _λ,µ ⁽¹⁾ (1) = V _λ,µ (1). However we have K _λ,µ ⁽¹⁾ (q) 6= V _λ,µ (q ² ) in general. For example if we take λ = (1, 0, 0) and µ = (1, 1, 1) we obtain K _λ,µ ⁽¹⁾ (q) = q ⁸ +2q ⁶ +2q ⁴ +q ² and V _λ,µ (q ² ) = q ¹⁰ +q ⁸ +2q ⁶ +q ⁴ +q ² . Consider ν in P n . For any standard tableau τ of shape ν and weight (1 ⁿ ), let ν ^′ be the standard tableau of shape ν ^′ and weight (1 ⁿ ) obtained by reflecting τ among the diagonal. Then one can verify that ch(τ ^′ ) = ⁿ⁽ⁿ⁻¹⁾ ₂ − ch(τ ) which by (8) implies the identity:

K _ν ^A

,(1

) (q) = q

K _ν,(1 ^A

) (q ⁻¹ ). (14) The following proposition will be useful in Section 4.

Proposition 3.3.2 Consider λ ∈ P n such that n ≥ |λ| Then we have

(i) : U _λ

_,(1

₎ (q) = q

^+n−|λ| u _λ,(1

₎ (q ⁻¹ ) and (ii) : u _λ

_,(1

₎ (q) = q

^+n−|λ| U _λ,(1

₎ (q ⁻¹ ).

Proof. By (ii) of Theorem 3.3.1 we have U _λ

_,(1

₎ (q) = q ^{n−|λ| 2} X

ν∈P

X

γ∈P

c ^ν _γ,λ

K _ν,(1 ^A

) (q). (15)

Since K _ν,(1 ^A

) (q) = 0 when |ν| 6= n, we can suppose that ν belongs to P b n = {ν ∈ P n , |ν| = n} in the above sum. For any ν in P b _n , we have c ^ν _γ,λ = 0 unless |γ| = n − |λ| . So we can suppose that γ belongs to P b n ⁽²⁾ = {γ ∈ P n ⁽²⁾ , |γ| ≤ n} in (15). The map Γ : ν 7−→ ν ^′ is an involution of P b n . Moreover Γ( P b n ⁽²⁾ ) = P b n ^(1,1) = {γ ∈ P n ^(1,1) , |γ | ≤ n}. Thus we can write

U _λ

_,(1

₎ (q) = q

n−|λ|

2 X

ν∈ P b

X

γ∈ P b

c ^ν _γ

,λ

K _ν ^A

,(1

) (q).

Now since c ^ν _γ

,λ

= c ^ν _γ,λ , we derive by (14)

U _λ

_,(1

) (q) = q

^{+ n−|λ| 2} X

ν ∈ P b

X

γ∈ P b

c ^ν _γ,λ K _ν,(1 ^A

) (q ⁻¹ ).

Finally the equality U _λ

_,(1

₎ (q) = q

^+n−|λ| u _λ,(1

₎ (q ⁻¹ ) is deduced from (i) of Theorem 3.3.1.

We obtain (ii) similarly.

Remark: By introducing for the root system A _n−1 generalized Kostka-Foulkes polynomials K _ν,R (q) where R = (R ₁ , ..., R _n ) is a sequence of rectangular partitions, Schilling and Warnarr [25] have proved the equality K _ν

_,R

(q) = q ^kRk K _ν,R

(q ⁻¹ ) where R ^′ = (R ^′ ₁ , ..., R ^′ _l ) and kRk = P

i<j |R i ∩ R _j | which can be considered as a generalization of (14). In [27], Shimozono and Zabrocki have also defined their polynomials K _λ,R ^♦ (q) when R is a sequence of rectangular partitions. By (12), the above proposition can also be regarded as a Corollary of Proposition 28 of [27].

4 Proof of Conjecture 2.5.2 when µ = (1, ..., 1)

4.1 The crystals B _Ξ

For any integer l ≥ 0, let B ^A (l) be the crystal graph of the irreducible finite dimensional U _q (sl _2n )- module of highest weight lΛ ₁ . In the sequel we choose to label the vertices of B ^A (1) by the letters of C n , that is we identify B ^A (1) with the crystal

1 → ¹ 2 · · · · → n − 1 ⁿ⁻¹ → n → ⁿ n ⁿ⁺¹ → n − 1 ⁿ⁺² → · · ·· → 2 ²ⁿ⁻¹ → 1.

Recall that the crystal graph B ^C (1) has been identified in 2.5 with

1 → ¹ 2 · · · · → n − 1 ⁿ⁻¹ → n → ⁿ n ⁿ⁻¹ → n − 1 ⁿ⁻² → · · ·· → 2 → ¹ 1.

Thus the crystals B ^A (1) and B ^C (1) have the same vertices. For any partition µ ∈ P n , set B _(µ) ^A = B ^A (µ ₁ ) ⊗ · · · ⊗ B ^A (µ _n ) and B _(µ) ^C = B ^C (µ ₁ Λ ₁ ) ⊗ · · · ⊗ B ^C (µ _n Λ ₁ ) (note that B ^C _(µ) 6= B _µ ^C defined in 2.5). Then B _(µ) ^A and B _(µ) ^C have also the same vertices. Nevertheless their crystal structure are distinct and their decompositions in connected components do not coincide.

Denote by H ^A the energy function associated to B ^A (l) ⊗ B ^A (k). Then for any b ₁ ⊗ b ₂ belonging to B ^A (l) ⊗ B ^A (k), we have H ^A (b 1 ⊗ b 2 ) = min(l, k) − m where m is the length of the shortest row of the semistandard tableau P ^A (b ₁ ⊗ b ₂ ) obtained by inserting the row b ₂ in the row b ₁ following the column bumping algorithm for semistandard tableaux. Given b = b ₁ ⊗ · · · ⊗ b _n ∈ B _(µ) ^A , set

H ^A (b) = X

1≤i<j≤n

H ^A (b _i ⊗ b ⁽ⁱ⁺¹⁾ _j )

where the vertices b ⁽ⁱ⁺¹⁾ _j are defined as in (7). In the sequel we need the following result due to Nakayashiki and Yamada:

Theorem 4.1.1 [20] Consider ν and µ two partitions of P n . Then K _ν,µ ^A

(q) = X

b∈G

q ^H

^(b)

where G ν is the set of highest weight vertices of weight ν in B _(µ) ^A .